A mesh "jumps" forward after parenting to an armature
Here is an example of an armature, edited in pose mode and, after this edit, made parent to the mesh (a cube):



This involves a 'shift' between pose and edit mode.

You could say 'there is no stable position for the mesh'.

To correct that, two possibilities: move the pose back to the edit position, or move the edit position to the pose.


Move the pose back to the edit position


To do that, simply use the pose menu, go to 'clear transform' and select 'all'. Doing this the pose will come back to the edit position:




Move the edit position to the pose


This time, we use the pose menu, go to pose then 'apply' (or CtrlA) and choose 'pose as rest pose'. Doing that, the edit position will be the pose position.



In both approach, the result is bones positions are the same in edit mode and pose mode (which is the best practice to define a 'stable' or 'reference' rest pose).

Depending on your need, you may have to parent mesh to armature again after that.

Additional indications: to avoid all that, adjust the bones in edit mode, not in pose mode. Also, if you scale the armature to the mesh, don't forget to apply this scale (CtrlA) before parenting to the mesh.